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April 12th, 2012, 11:01 PM  #1 
Member Joined: Apr 2012 Posts: 92 Thanks: 0  Inequalities, number theory
Hello all, I have a question and it may seem a little weird that I should ask it, but I really am confused. You see, I have just completed a topic on inequalities and was looking at number theory and found that the things I had learned in inequalities did not apply here, or so it seems. Find the sum of n terms of the GP 3+4.5+6.75+... How large must n be so that this sum is greater than 6000? I was able to identify common ratio r to be 1.5 Therefore, Sn>6000 Sn = 3(1(1.5)^n) / 11.5 >6000 3(1(1.5)^n) / 0.5 > 6000 6(1(1.5)^n)>6000 Because I cannot being over a negative with an inequality present, 6((1.5)^n 1) > 6000 ((1.5)^n 1) > 1000 And so on till I got n> log1.5(1001) n> 17.03, Since I cannot round down, n>18 ( Answer ) Was my working right? The answer was n>=18, can anyone tell me why? 6((1.5)^n 1) > 6000 ((1.5)^n 1) > 1000 This is the part where I'm confused. I vaguely remember that this is how you should approach this problem but what I learned in inequalities told me to in this case, square both sides since LHS and RHS are positive. OHHH actually I just remembered. I can bring over 6 because it is definitely positive right? 
April 12th, 2012, 11:24 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Inequalities, number theory
I would write: You use a weak inequality because n can be 18 or greater. Your working was correct, you just needed a weak inequality at the end. 
April 13th, 2012, 01:23 AM  #3 
Member Joined: Apr 2012 Posts: 92 Thanks: 0  Re: Inequalities, number theory
How did you get n1 and what is a weak inequality? Thank you!

April 13th, 2012, 05:06 AM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs  Re: Inequalities, number theory
I let the summation go from k = 0 to k = n  1, which is n terms. A strict inequality is strictly less than or greater than, while a weak inequality is less than or equal to or greater than or equal to. 

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